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A084950
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A factorial triangle, with row sums = a(n) of A001040.
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3
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1, 1, 2, 1, 6, 4, 24, 18, 1, 120, 96, 9, 720, 600, 72, 1, 5040, 4320, 600, 16, 40320, 35280, 5400, 200, 1, 362880, 322560, 52920, 2400, 25, 3628800, 3265920, 564480, 29400, 450, 1, 39916800, 36288000, 6531840, 376320, 7350, 36, 479001600, 439084800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Conjecture: also coefficient triangle of the denominators of the (n-th convergents to) the Continued Fraction w/(1+w/(2+w/3+w/.. This C.F. converges to 0.697774657964.. = BesselI[1,2]/BesselI[0,2] for w=1. [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010]
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FORMULA
| Given a flush left array, (r, c) of row (n+1) = (n+1)*(r-1, c) + (r-2, c-1); (i.e. multiply each term of n-th row by (n+1) and add the first term on the NW diagonal; getting the terms in row n+1.
(n-m)!/m!*binomial(n-m,m) [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010]
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EXAMPLE
| By rows, the triangle is:
0. 1
1. 1
2. 2 1
3. 6 4
4. 24 18 1
5. 120 96 9
6. 720 600 72 1
7. 5040 4320 600 16
...
E.g. to get row 7, multiply each term of row 6 by 7, then add the term NW of term in row 6: 5040 = (7)(720); 4320 = (7)(600) + 20; 600 = (7)(72) + 96; 16 = (7)(1) + 9. Thus row 7 = 5040 4320 600 16 with a sum of 9976 = a(7) of A001040.
The denominator of w/(1 + w/(2 + w/(3 + w/(4 + w/5)))) equals 120 + 96w + 9w^2. [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010]
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MATHEMATICA
| Table[CoefficientList[Denominator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k, {k, 1, n}]]]], w], {n, 16}]; or equivalently Table[( (n-m)!*Binomial[n-m, m] )/m! , {n, 0, 15}, {m, 0, Floor[n/2]}] [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010]
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CROSSREFS
| Cf. A001040.
Cf. A180047, A180048, A001849 [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010]
Cf. A180047, A180048, A001849 [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010]
Sequence in context: A005299 A185586 A128728 * A180317 A066654 A145960
Adjacent sequences: A084947 A084948 A084949 * A084951 A084952 A084953
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KEYWORD
| tabf,nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2003
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EXTENSIONS
| Rows 12 to 17 added based on formula. Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2010
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